Tensor Networks

- Fundamentals and graphical representation of tensor networks
- Mathematical approximation theory
- Backpropagation through tensor network operations
- Simulating strongly correlated quantum systems and digital quantum computers
- Probability distribution sampling using tensor networks

After successful completion of this module, students are familiar with the mathematical formalism and graphical notation for tensor networks. They can assess and apply tensor network approaches for approximating high-dimensional data. They understand why tensor network methods are suitable for simulating strongly correlated quantum systems, and are familiar with corresponding algorithms.

• MA0901 Linear Algebra for Informatics
• MA0902 Analysis for Informatics
• IN0018 Discrete Probability Theory
• Knowledge about quantum mechanics or computing helpful (but not a formal prerequisite)

The whiteboard lectures convey the mathematical formalism and graphical representation of tensor network methods and algorithm in-depth. The accompanying exercises for individual study deepen the understanding of the topics explained in the lecture, and foster the creative application of the learnt techniques.

Whiteboard or digital tablet, slides

W. Hackbusch, S. Kühn: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706 (2009)

M. Espig, W. Hackbusch, S. Handschuh, R. Schneider: Optimization problems in contracted tensor networks. Comput. Visual Sci. 14, 271 (2011)

U. Schollwöck: The density-matrix renormalization group in the age of matrix product states. Annals of Physics 326, 96 (2011)

R. Orús: Tensor networks for complex quantum systems. Nature Reviews Physics 1, 538 (2019)

J. Haegeman, Ch. Lubich, I. Oseledets, B. Vandereycken, F. Verstraete: Unifying time evolution and optimization with matrix product states. Phys. Rev. B 94, 165116 (2016)

L. Vanderstraeten, J. Haegeman, F. Verstraete: Tangent-space methods for uniform matrix product states. SciPost Phys. Lect. Notes 7 (2019)

H.-J. Liao, J.-G. Liu, L. Wang, T. Xiang: Differentiable programming tensor networks. Phys. Rev. X 9, 031041 (2019)

The assessment is by means of a written exam of 90 minutes, or a 20 minute oral examination. Problems on the underlying approximation theory test whether students can apply the acquired knowledge and methods to a new domain. Problems related to the graphical representation may ask students to express an algorithm stated in matrix notation into a graphical form. The exam assesses the knowledge of tensor network methods for simulating quantum systems for example by asking them to tailor an algorithm for a given quantum Hamiltonian. Reaching a certain fraction of the maximum number of homework points can be rewarded by a grade step bonus (“eine Notenstufe”) in the evaluation of the exam. The exact details are timely announced at the beginning of the lecture.